Conditional and unconditional heteroscedasticity in the market model

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FACULTY WORKING
PAPER NO. 1218

Conditional and Unconditional Heteroscedasticity
in the Market Model

Anil Bera
Edward Bubnys
Hun Park

HE

i986

College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois, Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO 1218
College of Commerce and Business Administration
University of Illinois at Ur bana-Champa i gn
January 1986

Conditional and Unconditional
He t e r os ceda s t i c i t y in the Market Model

Anil Bera, Assistant Professor
Department of Economics

Edward Bubnys
Memphis State University

Hun Park, Assistant Professor

Department of Finance

This research is partly supported by the Investors in Business Education
at the University of Illinois

Conditional and Unconditional Heteroscedasticity
in the Market Model

Abstract

Unlike previous studies, this paper tests for both conditional and
unconditional heteroscedasticities in the market model, and attempts
to provide an alternative estimation of betas based on the autoregres-
sive conditional heteroscedastic (ARCH) model introduced by Engle.
Using the monthly stock rate of return data secured from the CRSP tape
for the period of 1976-1983, it is shown that conditional heterosce-
dasticity is more widespread than unconditional heteroscedasticity,
suggesting the necessity of the model refinements taking the condi-
tional heteroscedasticity into account. In addition, the efficiency
of the market model coefficients is markedly improved across all firms
in the sample through the ARCH technique.

Conditional and Unconditional Heteroscedasticity
in the Market Model

I. Introduction

Over the past two decades, the capital asset pricing model (CAPM)
has heen more widely used in finance than any other model. Inherent
in the CAPM is that capital assets, under some simple assumptions, are
held as functions of the expected means and variances of the rates of
return, and that investors pay only for the systematic risks of capi-
tal assets which are measured by the relationship between the return
on individual assets and the return on the market (e.g., the single-
index market model of Sharpe [14]). However, a number of studies have
raised questions on the validity of the market model to estimate the
systematic risk of financial assets using the ordinary least squares
(OLS) technique. In particular, a significant portion of the previous
studies have focused on the variance structure of the market model
(see for example [1], [2], [4], [5], [6], [8], [91, [12] and [13]).

In the traditional approach, when the mean is assumed to follow a
standard Linear regression, the variance is constrained to be constant
over time. However, many studies have provided strong evidence of
heteroscedasticitv for a number of common stocks not only in the U.S.
market but in the Canadian and European markets and attributed in gen-
eral to model misspecif ication either by omitted variables or through
structural changes. Giaccotto and Ali [10] used two classes of opti-
mal nonparametric distribution-free tests for heteroscedasticity and
applied them to the market model. They found the assumption of
homoscedasticity to be untenable for the majority of stocks analyzed.

-2-

Lehmann and Uarga fll] took issue with Giacotto and Ali's use of
recursive residuals in rank tests in order to provide some information
concerning the stochastic properties of market model regressions.

Nevertheless, most of the previous studies have heen limited to
investigating the existence of heteroscedasticity in the market model
using different statistical tests and rarelv considered estimation of
betas taking heteroscedasticity explicitly into account. Furthermore,
they considered only unconditional variances of the disturbance term,
ignoring the possible changes in conditional variances. Some studies
attempted to find a form of heteroscedasticity in an ad-hoc fashion.
The predominant approach was to introduce an exogeneous variable (the
market return in most cases) which may predict the variance. As
pointed out by Engle [7], this conventional solution to the problem
requires a specification of the causes of the changing variance in an
ad-hoc fashion rather than recognizing that both means and variances
conditional on the information available may "jointly evolve over time.

The purpose of this study is to test for both conditional and
unconditional heteroscedasticities, and provide an alternative estima-
tion of betas based on the autoregressive conditional heteroscedastic
(ARCH hereafter) model, introduced by Engle [7]. The ARCH process is
characterized by mean zero, seriallv uncorrelated processes with non-
constant variances conditional on the past but constant unconditional
variances. This study is different from the earlier ones in some
important aspects. First, using the ARCH model, this study provides
a more realistic measure of betas when the underlying conditional

-3-

variance mav change over Lime and is predicted by the past forecast
errors rather than making conventional assumptions about the distur-
bances. Second, it provides more efficient estimators using the maxi-
mum likelihood method. Third, it does not employ an arbitrary
exogenous variable such as the market return to explain heterosce-
dasticity. Lastly, by the nature of the ARCH process, the effects of
omitted variables from the estimated model, and a portion of non-
normality of the regression disturbance terms, may be picked up.

Section IT describes the model and section III presents empirical
results. Section IV contains a brief summary.

II. Model

Consider the single index market model

R. = a. + 8 .R + E. (1)

it l l mt it

where R. and R , are the random return on security i and the random
it mt J

market return, respectively, in period t; a. and 8. are the regression

parameters of security i; E. is the random disturbance term with

it

E(E. ) = 0 for all i and t. The parameter 8. measures the systematic
it i J

risk of security i and is defined as Cov(R. , R )/Var(R ). One of the

l m m

many assumptions that model (1) is based on is that the disturbances
are homoscedastic.

First, we check for unconditional heteroscedasticity using the
White [15] test which does not assume anv specific form of the hetero-
scedast icitv. Second, we test for conditional heteroscedasticity and

-4-

simultaneouslv attempt to reestlmate the market model with an ARCH
specification.

For ease of exposition, let us put model (1) in a general linear
regression framework in matrix notation:

Y = X 8 + E (2)

(Txl)(Txk)(kxl)(Txl)

with k=2. Let 8 be the OLS estimator. If the E's are homoscedastic ,
then a consistent estimator of the variance-covariance matrix of 8 is
given by

VjCS) = a2(X'X)-1 (3)

where a = E'E/T-k and E's are the OLS residuals.

In the presence of heteroscedescity , a consistent estimator of 8
wi 1 1 be

V2(6) = (X'X)"1(X'EX)(X'X)"1 (4)

~ 2 ~ 2 ~2

where E = diag (E , E-, . . . , ET).

Under homosecdasticity , these two estimates V.(8) and V„C8), will
converge to the same limit. However, in case of possible heterosce-
dasticity, V.(8) is inconsistent and these two estimators will have
different limits. White's procedure is based on examining the sta-
tistical significance of the limits of the k(k+l)/2 distinct elements of

the matrix V = V (8 ) - V (8). Under the assumption of homoscedast icity ,

2
the test statistic is asymptotically distributed as x with k(k+l)/2

degrees of freedom. One attractive feature of this test is that it

-5-

does not depend on a specific form of heteroscedasticity. Assuming

that the disturbances are homokurtic, the test statistic can be

2 ?
calculated as T*R where R is the coefficient of determination

obtained by regressing the square of the residuals on a constant,

- 2
R and R~ . The test statistic will have 2 degrees of freedom (one
mt mt

less because of the presence of an intercept term in the regression).

Following Engle [7], we assume that the conditional heterosce-
dasticity of the market model can be represented by a first-order ARCH
process (suppressing the suffix i) as:

V(VW =°t -*o + *i Et-i • <5)

where Yn > 0, anH 0 < y < 1.

This tvpe of conditional heteroscedasticity has some intuitive
appeal since it does not depend on some arbitrarv exogenous variables
and can be viewed as some average of the square of the past distur-
bances. When Yi = ^» we have conditional homoscedasticity. It can be
easily shown that unconditionally E(E ) = 0, V(E ) = Y /(1~Y,) and
E(E E ,) = 0 for t * t' (see Engle [7] for details). Since we do not
have anv lagged dependent variable in our model, V. (8 ) - V„(B) will
have the same probability limit. Therefore, White's procedure will
test only for unconditional heteroscedasticity, while the test for
Y, = 0 in (5) will provide a test for conditional heteroscedasticity.
It is also important to note that under the ARCH model, the E 's are
uncorrelated but dependent, which is a property of non-normal dis-
tributions .

The log-likelihood function assuming normality is given hv

-6-

T
L = Z (a -\ log a1 - \ Z2J°h (6)

t = l

where a is a constant term and E = R - a - 8 R . Using Ouandt's

t t mt

subroutine GRADX of his numerical optimization program 0Q0PT3, we
maximize the function L and obtain estimates of a, 3, Yn and y..
These estimates of a and 8 are compared with the OLS estimates.
Significance of y, will indicate the presence of conditional
heteroscedasticity. This is tested using t-statistics.

We also test for normality of the disturbances of the OLS and ARCH
models. The test statistic is calculated as:

m r(skewness) (kurtosis-3) ,
T [ - + _ ].

Under normality this test statistic asymptotically follows a central

2
X with 2 degrees of freedom (see Bera and Jarque [3] for detaiLs).

III. Empirical Results

Monthly stock rate of return data was secured from the CRSP tapes
for the period 1976-83 (96 months) for 35 randomly chosen firms with-
out missing values. Market model regressions using the OLS and ARCH
methods were run for each firm. The homoscedasticity and normality
tests statistics also were calculated. The results based on the OLS
regression and the ARCH model are shown in Tables 1 and 2, respec-
tively.

First, from the results on the White test we can observe that only
for ten firms the test statistics are significant at 1% and 5% levels.
This indicates that the unconditional heteroscedasticity may not be
very important. However, all of the normality test statistics are

-7-

significant indicating the presence of strong non-normality. The OLS
results exhibit the familiar strong significance of the beta coef-
ficient, with 33 of the market models having beta t-values significant
at the 1% level. The intercept a terms are, for the most part, small
and insignificant.

The importance of conditional heteroscedasticity and thus the
appropriateness of the ARCH model can be judged from the significance
of 26 y, values in Table 2. It should be noted, however, that the
significance of y . (a test of conditional heteroscedasticitv) in Table
2 has no relation with the earlier test for unconditional heterosce-
dasticitv in Table 1. This is not surprising since they test two
completely different hypotheses. For the ARCH model also, the 33
betas are significant and the significance of several of the inter-
cepts is enhanced. Of particular interest is the change in beta as a
result of the use of ARCH. Fourteen of the 33 firms with significant
OLS betas have increased betas when ARCH is used; nineteen others have
beta reductions. However, most of the changes are small. Nineteen
ARCH model betas change no more than 4 percent up or down from their
OLS counterparts. Six firms' betas increase by 5% or more when ARCH
is used, while eight betas decrease by at least 5%. Only five firms
(14.3% of the sample) have betas change by 10% or more.

The efficiency of the market model coefficients is markedly im-
proved across all firms. The t-values for all the ARCH model betas
are higher than those for the OLS model due to the lower standard
errors of the coefficients. Thus it appears that incorporation of a

-8-

conditional heteroscedastic component results in greater efficiency,
though affecting the parameter estimates themselves slightly. In sum,
the differences between the OLS and ARCH results reveal the importance
of taking account of conditional heteroscedasticity in the market
model.

Lastly, the results on normality tests indicate strong non-
normality of the disturbance terms in the ARCH model. This is what we
should expect because, as pointed out earlier, the ARCH errors are
inherently non-normal or mutually dependent. The ARCH procedure takes
account of this type of non-normality or the dependent structure,
while the OLS procedure (results of Table 1) neglects it. The gain is
reflected in the improved standard errors of a and B in Table 2.

IV. Summary

A number of studies have shown that heteroscedasticity in the
market model is widespread. However, the previous studies have been
limited to investigating the existence of heteroscedasticity using
different statistical tests and rarely considered estimation of betas
taking heteroscedasticity explicitly into account. Furthermore, they
considered only unconditional heteroscedasticity, ignoring conditional
heteroscedasticity. This paper tests for both conditional and uncon-
ditional heteroscedast icities and provides an alternative estimation
of betas based on the autoregressive conditional heteroscedastic
(ARCH) model introduced by Engle. Using the monthly stock rate of
return data secured from the CRSP tape for the period of 1976-1983, it
is shown that conditional heteroscedasticity is more widespread than
unconditional heteroscedasticity, suggesting the necessity of the

-9-

model refinements taking the conditional heteroscedasticity into

account. In addition, the efficiency of the market model coefficients

is markedly improved across all firms in the sample through the ARCH
technique.

-10-

Ref erences

ll] Barone-Adesi, G. and P. Talwai. "Market Models and Heteroscedasti-

city of Residual Security Returns." Journal of Business and Economic
Statistics, 1 (1983), pp. 163-168.

[2] Belkaoui, A. "Canadian Evidence of Heteroscedasticity in the Market
Model." Journal of Finance, 32, (September 1977), pp. 1320-1324.

[3] Bera. A. K. and C. M. Jarque. "Model Specification Tests: A

Simultaneous Approach." Journal of Econometrics, 20 (1982), pp.
59-82.

[4] Bey, R. P. and G. E. Pinches. "Additional Evidence of Heterosce-
dasticity in the Market Model," Journal of Financial and Quan-
titative Analysis, 15, (June, 1980), pp. 299-322.

[5] Brown, S. J. "Heteroscedasticity in the Market Model: A Comment."
Journal of Business, 50, (January 1977), pp. 80-83.

[6] Brown, K. C. , Lockwood, L. J. and S. L. Lummer. "An Examination of
Event Dependency and Structural Change in Security Pricing Models."
Journal of Financial and Quantitative Analysis, 20, (September 1985),
pp. 315-334.

[7] Engle, R. F. "Autoregressive Conditional Heteroscedasticity with

Estimates of the Variance of United Kingdom Inflation," Economet rica ,
Vol. 5, No. 4, (July 1982), pp. 987-1007.

[8] Fabozzi, F. J. and J. C. Francis. "Heteroscedasticity in the Single
Index Market Model," Journal of Economics and Business, 32, (Spring
1980), pp. 243-248.

[9] Fisher, L. and J. Kamin. "Forecasting Systematic Risk: Estimates
of "Raw" Beta that Take Account of the Tendency of Beta to Change
and the Heteroskedast icity of Residual Returns," Journal of
Financial and Quantitative Analysis, 20, (June 1985), pp. 127-150.

10] Giaccotto, C. and M. Ali. "Optimal Distribution-Free Tests and

Further Evidence of Heteroscedasticity in the Market Model." Journal
of Finance, 37, (December 1982), pp. 1249-1258.

11] Lehmann, B. and A. Warga. "Optimal Distribution-Free Tests and
Further Evidence of Heteroscedasticity in the Market Model: A
Comment." Journal of Finance, 40, (June 1985), pp. 603-605.

12] Martin, J. D. and R. C. Klemkosky. "Evidence of Heteroscedasticity
in the Market Model." Journal of Business, 48, (January 1975), pp.
81-86.

-li-

tis] McDonald, B. and M. Morris. "The Existence of Heteroscedastici ty and
its Effect on Estimates of the Market Model Parameters," The Journal
of Financial Research, Vol. VI, No. 2, (Summer 1983), pp. 115-126-

[14] Sharpe, W. F. "A Simplified Model for Portfolio Analysis." Manage-
ment Science, 9, (January 1963), pp. 277-293.

[15] White, H. "A Heteroskedasticity — Consistent Covariance Matrix

Estimator and a Direct Test for Heteroskedasticity," Econometrica,
48, (1980), pp. 817-838.

D/315

Table 1

Results Based on the Ordinary Least Squares Regression
( t-statistics are in parentheses)

Homoscedasticity*

Firm No.

a

8

(White test)

NormaliLv*

1

-.002
(-.026)

1.22a
(7.70)

0.634

36.33a

2

.003
(0.29)

1.06a
(4.82)

3.005

22.40a

3

-.003
(-0.66)

1.02a
(8.72)

1.152

19. 29a

4

.008b
(2.10)

0.69a
(8.35)

3.562

21.03a

5

.002
(0.18)

1.75a
(6.72)

8.774b

14.72a

6

.021b
(1.71)

0.07
(0.24)

0.115

14.473

7

.009
(1.02)

-0.14
(-0.69)

8.304b

22.72a

8

.012
(1.58)

0.75a
(4.44)

9.053b

33.81a

9

.021
(1.43)

1.73a
(5.31)

0.557

20.793

10

.002
(0.33)

1.02a
(6.81)

2.803

23.75a

1 1

-.001

1.05a

16.4353

21.083

(-0.15)

(7.32)

9.05b

12

.015

1.87a

0.701

(1.64)

(8.86)

13

.010
(1.34)

0.99a
(5.88)

6.682b

42.68a

14

.002
(0.23)

1.07a
(6.95)

1.123

16. 31a

15

.010
(1.03)

1.28a
(5.69)

4.848

20.39a

16

.016
(1.38)

0.77a
(3.04)

2.131

33.26a

17

-.006
(-0.85)

1.26a
(8.36)

10.589a

47.57a

*x2

test statistics.

Significant at the VI level; the critical value for t-statistics

2.37 and the critical value for x statistics = 9.21.

Significant at the 5% level; the critical value for t-statistics

1.66 and the critical value for x statistics = 5.99.

Table 1 (con't. )

Results Based on the Ordinary Least Squares Regression
( t-statistics are in parentheses)

Homoscedasticitv*

Firm No.

a

B

(White test)

Normal i ty

18

-.003
(-0.37)

0.96a
(5.92)

3.907

40.c>7a

19

.006

1.08a

0.816

21.74a

(0.67)

(5.18)

7.73b

20

.010

0.76a

0.701

(1.20)

(4.25)
0.97a

21

.001

4.080

32.42a

(0.18)

(6.54)

22

.003
(0.47)

0.88a
(5.74)

2.390

44.86a

23

.009
(1.08)

1.95a

(10.41)

13.8343

33.27a

24

.009
(1.31)

0.83a
(5.13)

11.539a

37.97a

25

-.001
(-0.08)

1.31a

(6.21)

2.688

33. 19a

26

.008
(1.25)

0.83a
(5.90)

9.907a

13.64a

27

.006
(1.00)

0.89a
(6.74)

5.126

25.66a

28

.002
(0.40)

0.58a
(4.55)

0.048

19.31a

29

.011
(0.68)

0.94a
(2.68)

4.0^0

36.113

30

-.003
(-0.41)

1.05a
(6.44)

4.906

34.26a

31

.024b
(2.30)

1.31a
(5.64)

3.629

17.40a

32

,011b

(1.77)

1.07a
(7.80)

4.704

29.44a

33

.009
(0.97)

1.1 5a
(5.31)

3.725

38.32a

34

.023a
(2.79)

1.27a
(6.81)

10.0033

18.213

35

.007
(0.76)

1.71a
(8.72)

2.803

30.633

*x2

test statistics.

a.

Significant at the L% level; the critical value for t-statistics =
2.37 and the critical value for x2 statistics = 9.21.

Significant at the 5% level; the critical value for t-statistics =
1.66 and the critical value for x2 statistics = 5.99.

Table 2

Results Based on the ARCH Model
( t-stattstics are in parentheses)

a

a

^

»

Normality

Firm No.

a

8

Y0

.003a

.305a

Test Statistics*

1

-.003

1.33a

33.37a

2

(-.61)
.004

(12.30)
0.95*

(5.90)
.007a

(2.18)
.172

22.56a

3

(0.69)
-.003

(6.46)
-0.98a

(7.09)
.002a

(1.47)
.142

19.l4a

4

(-0.095)
.008a

(12.51)
0.64a

(7.62)
,008a

(1.59)
.275a

20.183

5

(3.61)

.002

(12.15)
1.77a

(6.68)
.010a

(2.40)
.204

14.493

6

(0.23)
.018b

(10.48)

-0.05

(6.85)

.oua

(1.61),
.166b

14.23a

7

(2.24)
.008

(-0.25)
-.216

(7.83)
.006a

(1.72)
.106

22.70a

8

(1.39),

.onb

(-1.59)
.784a

(8.14)
.003a

(1.50),
.310b

33.33a

9

(2.23)
.023a

(7.03)
1.74a

(6.11)
.017a

(2.29)

.077

20.7 53

10

(2.42)
.005

(8.14)
1.02a

(8.80)
.003a

(1.30),

.241b

23.82a

11

(1.10)
-.003

(10.44)
1.18a

(6.63)
.003a

(1.90),
.205b

15.423

12

(-0.82)
.016a

(12.69)
1.90a

(7.63)
.007a

(2.11)
.118

9.09b

13

(2.51)
.009

(13.64)
0.98

(7.98)
.004a

(1.44)
.182b

42.87a

14

(1.76)

.001

(8.67)
1.03a

(7.12)
.0033

(1.84),
.138b

16.113

15

(0.30),

.onb

(10.11)
1.55a

(8.22)
.005a

(1.83)
.452a

19.38a

16

(1.79)
.017a

(11.34)
0.72a

(6.01)
.008a

(2.99),

.254b

33.37a

17

(2.38)
-.007

(4.48)
1.23*

(6.77)
.003a

(2.15)
.149

47.703

(-1.62)

(12.44)

(6.92)

(1.42)

test statistics.

Significant at the 1% level; the critical values for t-statistics
2.37 and the critical value for x statistics = 9.21.

Significant at the 5% level;„the critical values for t-statistics
1.66 and the critical value for x statistics = 5.99.

Table 2 (cont'd. )

Results Based on the ARCH Model
( t-stat ist ics are in parentheses)

^

»

Normality

Firm N'o.

a

8

Y0
.004a

Yl
.142b

Test Statistics*

18

-.003

0.92a

40.613

(-0.70)

(8.45)
1.09a

(7.74)

(1.69),
.273b

19

.004

.006

21.86a

(0.75)

(8.11)

(6.91)

(2.18)

7.79b

20

.007

0.753

.003a

.392a

(1.45)

(6.85)

(6.04)

(2.65),

.137h

21

.0008

0.95a

.003a

32.61a

(0.17)

(9.65)
0.89

(7.88)

(1.72),
.243b

22

.003

.003a

44.96a

(0.70)

(8.98)

(5.68)

(1.69),

.221b

23

.006

1.80a

.005a

33.43a

(1.17)

(14.57)

(7.17)

(2.20)

.224h

24

.011b

0.84a

.no3a

37.94a

(2.29)

(7.86)
1.29a

(6.55)

(1.96)
.183

25

-.00003

.006

32.39a

(-0.005)

(9.39)

(7.25)

(1.81)

26

.004

0.89a

.002

.506a

14.08a

(1.14)

(10.18)

(6.10)

(3.04)
.357b

27

.006

0.82a

.002a

25.69a

(1.62)

(9.79)
0.68

(6.04)

(2.19),

.244b

28

-.0009

.002a

16.43a

(-0.03)

(8.15)

(7.42)

(2.33)u
.138b

29

.006

0.90a

.018a

36.48a

(0.60)

(3.87)

(7.88)

(1.71)

30

-.002

1.09a

.004a

.117

34.45a

31

(-0.33)
.024a

(10.07)
1.20a

(8.12)
.008

(1.62),

.151b

16.783

(3.47)

(7.72)

(7.98)

(1.85)

32

.010a

0.95a

.002a

.343a

27.83a

(2.55)

(11.09)

(6.61)

(2.76)

.210

33

.008

1.16a

.006a

38.163

(1.20)

(8.03)
1.15a

(6.55)

(1.66),
.243b

34

.022a

.005a

18.173

(4.12)

(9.39)

(6.72)

(1.92),
.316b

35

.004

1.65a

.005a

29.86a

(0.77)

(12.64)

(6.13)

(2.34)

test statistics.

Significant at the 1% level; the critical values
2.37 and the critical value for x statistics = 9.21.

for t-statistics =

Significant at the 5% level; the critical values
1.66 and the critical value for x~ statistics = 5.99.

for t-statistics =

HECKMAN

BINDERY INC

JUN95

iv^J T.fV.^ N MANCHESTER.